On a general class of gamma based copulas

: A large family of copulas with gamma components is examined, and interesting submodels are defined and analyzed. Parameter estimation is demonstrated for some of these submodels. A brief discussion of higher-dimensional versions is included.


Introduction
Arnold and Ng [3] introduced a bivariate second kind beta, or beta(2), distribution involving 8 independent random variables with gamma distributions (subsequently such random variables will be referred to as gamma components). It was identi ed as the most general bivariate model whose marginals are ratios of sums of independent gamma variables. The model involves independent components U , U , ..., U with U j ∼ Γ(α j , ), j = , , ..., . The two-dimensional random vector (X, Y) is then de ned by This de nes an -parameter family of bivariate distributions with beta(2) marginal distributions. If (X, Y) is de ned as in (1) then we write: (X, Y) ∼ BB( )(α), to be read as (X, Y) has a bivariate second kind beta distribution with parameter vector α. Each α i can assume any positive real value, so that the parameter space of the model is ( , ∞) . A corresponding family of bivariate distributions with beta marginals of the rst or usual kind is obtained from (1) by de ning (V , V ) = (X/( + X), Y /( + Y)).
Why there are U j 's, and where they are located in the model (1), may require some explanation. There are four locations where a particular U j may be placed. (1) In the numerator of X. (2) In the denominator of X. (3) In the numerator of Y and (4) In the denominator of Y. The variables U , U , U and U appear only once, and each one of them appears in only one of the four possible locations. A variable U j cannot appear in both the numerator and denominator of X, nor of Y, since otherwise the independence of numerators and denominators, required for beta(2) marginals, would be destroyed. U appears in the numerator of both X and Y. U appears in the denominator of both X and Y. U appears in the numerator of X and in the denominator of Y, while U appears in the denominator of X and in the numerator of Y No U j can appear in or in of the possible locations, since that would destroy the required independence of at least one numerator and its corresponding denominator. If an additional independent gamma variable is introduced in one or two permissible locations in (1) then it can be combined with one of the existing U j 's and no enrichment of the model will result. Thus for example, if U is added to both numerators, then U + U will continue to play the role of U with an adjusted shape parameter α + α .
We adopt the convention that a random variable with a Γ(α, ) distribution with α = will be de ned to be a random variable that is degenerate at . By setting some of the α j 's in the Arnold-Ng model (1) equal to zero, simpli ed submodels (some of which have been discussed in the literature) will be obtained. Note that after setting certain α j 's equal to zero, we must retain α + α + α > , α + α + α > , α + α + α > , and α + α + α > , in order to continue to have beta(2) marginal distributions.
While this general bivariate beta model, and particularly its submodels, have demonstrated exibility and usefulness in practice, the focus of this paper is on a speci c class of submodels of the beta(1) containing only copulas, that is, distributions with uniform marginals. This paper is organized as follows. In Section 2, we construct this speci c class of copulas by limiting the parameter space and discuss how further limitations of the parameter space can produce several familiar copula models discussed in the literature. In Section 3, natural symmetries of the copulas are considered. Section 4 discusses higher-dimensional versions. We demonstrate some examples of parameter estimation in Section 5, and nish with concluding remarks in Section 6.

Copulas
Our gamma based copulas are obtained by setting the values of the parameters in the Arnold-Ng(8) bivariate beta distribution so that the marginals are Uniform( , ).
Recall that the Arnold-Ng -parameter bivariate beta model is of the form where U , U , U , ..., U are independent random variables with U i ∼ Γ(α i , ), i = , , ..., .
In order to have uniform marginals the α i 's must satisfy: and where all the α i 's are non-negative (and necessarily ≤ ). We may rewrite these constraints in the following form: so that the parameter space, the set of permissible values for (α , α , α , α ), may be de ned by: It is convenient to use notation BB(i , i , ..., i k ) to denote the model obtained from (2) by setting certain α i 's equal to 0 and retaining only α i , α i , ..., α i k in the model. So, for α = α = and α = α = , this model reduces to the co-monotone copula (BB(5,6)), that is, V = V . Likewise, if α = α = and α = α = , this model reduces to the counter-monotone copula (BB (7,8)), that is, V = − V . Also, if α = α = α = α = (BB(1,2,3,4)), then V is independent of V . Therefore, this model encompasses a continuous space of distributions with the co-monotone and counter-monotone copulas, also known as the Fréchet-Hoe ding bounds, and the independent copula as boundary points. Throughout this paper, correlations for this model or its submodels are expressed in terms of Spearman's Rank Correlation and are obtained by simulation when they cannot be determined analytically.
Submodels of this family are of particular interest. For example, the simplest is given when α = α, and α = α = α = , so that it takes the form: This family has positive correlations ranging from to .
(discussed in more detail later in the section), and the correlation has a monotone relationship with the parameter α. At α = , this model reduces to the independent case. A larger submodel, the Magnussen model, is obtained by setting only α = α = (see [5]). It thus is of the form: where and where all the α i 's are non-negative (and necessarily ≤ ). This is a two parameter family of copulas parameterized by α , α ∈ ( , ). The other four α i 's are determined by (11)-(14). This model exhibits a full range of positive correlations, with the independent copula and the co-monotone copula appearing at the two extremes of the parameter space. The Arnold-Ng parameter model (introduced in [3]) is obtained by setting α = α = α = , and is therefore denoted by BB (1,2,6,7,8). The corresponding family of copulas includes those of the form: where and where all the α i 's are non-negative (and necessarily ≤ ). This is a one parameter family of copulas parameterized by α ∈ ( , ). The other four α i 's are determined by (16)-(19).
We will denote these models by AN(5A)=BB(1,2,6,7,8), AN(5B)=BB (3,4,5,7,8), AN(5C)=BB(2,3,5,6,7), and AN(5D)=BB (1,4,5,6,8). For these to be copulas we must impose the following constraints: For AN(5A): For AN(5B) For AN(5C) For AN(5D) So we have four one parameter families of copulas. But there are more one parameter submodels that can be constructed by setting three of the ve parameters to zero and applying the appropriate constraints. Four of the = choices must be excluded as they would produce zeros in one of the numerators or denominators in (1), and four additional choices all represent the same family as the following example shows. Take the BB(4,5,6,7,8) model for which we need to impose the following constraints: We can take α = α ∈ ( , ). Then α = − α, α = − α, α = α, and, necessarily, α = , so our model is equivalent to the BB(5,6,7,8) with constraints. But how can one pick among all these models, and what properties do they have?
Let us look at the Olkin-Liu BB [7] model: For this to be a copula, we need α = α = α = . Thus we have a -dimensional family of copulas. Three related copulas are obtainable by re ection about / . Clearly, when α = for the simple family (9), it reduces to a re ected version of (40), and when the Magnussen parameters satisfy α = − α , it also reduces to (a possibly re ected version of) this model. As a copula, the Olkin-Liu BB model is also known as the Ali-Mikhail-Haq copula [1], a distribution that tends to appear in a vast array of copula families, including many of the Archimedean type (It should be noted that while this paper refers to the "Ali-Mikhail-Haq" copula multiple times, the copula that is being referenced is only one speci c member of an entire family of copulas which is not a subset of the present family.) [6]. Let us return to consider the Arnold-Ng(5A) model of the form: where α + α = , and (44) where all the α i 's are non-negative (and necessarily ≤ ). This is a one parameter family of copulas parameterized by α ∈ ( , ). The other four α i 's are determined by (42)-(45). So we have α = α = α = α ∈ ( , ) and α = α = − α. A plot of the correlation of the AN(5A) model, as a function of α, is given in Figure 1.
One immediate observation about the AN(5A) model is that its range of correlations is not full, that is, it is a proper subset of [− , ]. While this may at rst thought appear to be a disappointing range of correlations, it should be noted that this family exhibits a di erent feature that may be desirable for many applications: it is a continuous collection of distributions ranging from the counter-monotone copula (having a correlation of -1) to an Ali-Mikhail-Haq copula, which favors strong upper tail (positive) dependence only, which coincides with the 0.478 correlation. Contour plots of the densities for various values of α are shown in Figure 2. It is applicable to phenomena that do not exhibit signi cant lower tail dependence, but do exhibit a trade-o between upper tail dependence and an overall negative correlation. For example, if X represents a citizen's government entitlements, and X represents the same citizen's overall wealth, then applying the AN(5A) to V = F X (X ) and V = F X (X ), α could be a measure of the government's plutocratic tendencies, since we might observe α nearer to for socially democratic governments, while α may be much closer to 1 for highly plutocratic governments.
Remark Among the avatars of the copula based on the Arnold-Ng parameter bivariate beta model, two, namely AN(5A) and AN(5B), contain the counter-monotone copula but not the co-monotone copula. In contrast AN(5C) and AN(5D) contain the co-monotone copula but not the counter-monotone copula.

Symmetry considerations
There are three symmetry conditions that are on occasion deemed to be desirable properties for modeling purposes. It is therefore of interest to identify which members of our general copula family exhibit such symmetry features. A copula will be said to be marginally symmetric if signi es equality of the distribution functions associated with the random variables. Clearly any BB copula with parameter vector α satisfying the constraints (3)-(6) will exhibit marginal symmetry since V and V have Uniform( , ) distributions. Radial symmetry requires more restrictions. In order for a copula to be radially symmetric it must be the . For this to be true the parameter vector α must be of the form where ≤ α + β ≤ . We thus have a two parameter subfamily of radially symmetric copulas.
For joint symmetry, even more is required, namely (V , V ) where ≤ α ≤ . We thus have a one parameter subfamily of jointly symmetric copulas.
There is another "symmetry" condition that can be considered. We will say that a copula is exchangeable . For this to be the case, the parameter vector of the copula must be of the form This is a three parameter family of copulas where the parameters satisfy the following constraints As examples, the AN(5C) and AN(5D) (among many others) are exchangeable.

On multivariate gamma based copulas
k-dimensional versions of the Arnold-Ng beta(2) distribution were mentioned in Arnold and Ghosh [2], in a context of copula models. First we consider the three dimensional case. It will then be evident how to deal with higher dimensions. A three dimensional beta(2) distribution will be one whose structure is of a form which involves independent gamma distributed U j 's. Thus, there are a total of parameters in the model where U j , j = , , ..., are independent variables with U j ∼ Γ(α j , ) for each j. The model can then be expressed in the following form.
We must then impose constraints to ensure that the associated trivariate beta distribution has uniform marginals.
The pattern for the dimensions of the parameter spaces of the multivariate copula models can now be recognized. The univariate model involves U's with two constraints, i.e., a − − = parameter model. The bivariate model involves U's with constraints, i.e., a − − = parameter model. The trivariate case involves U's with constraints, i.e., a − − = parameter model, and, in general, the k-dimensional model involves k − U's with k constraints.
Use of fully parameterized k-dimensional copulas would almost never be recommended. Instead simplied sub-models, obtained by setting many of the α's equal to zero, can be expected to be adequate for many data sets.

Parameter estimation
In general, densities are not available in analytic form for the gamma-based copulas under discussion, except for special cases, such as the Ali-Mikhail-Haq case. So, if a sample is available from the bivariate gamma based copula then maximum likelihood is unavailable for parameter estimation. What we do have available are relatively simple expressions for mixed moments of the coordinate random variables. In principle, then, we could choose several sample mixed moments, equate them to their expectations and solve the resulting equations for the α i parameters. This, in many cases, will prove to be a non-trivial exercise. However, it will typically be feasible for simpli ed sub-models involving only a few of the α i 's. With fewer equations to deal with, the method of moments approach is often not di cult to implement.

Example 1. As a very simple example consider a model in which α
In this one parameter family of copulas we can denote α by α and then we have α = α and α = α = − α. In this case, we can verify that E(V V ) = (α + )/ .
If we de ne M V V = ( /n) n i= V i V i , then a moment based estimate of α will be

Example 2.
Suppose that (V , V ) is of the AN(5A) type. While it may be tempting to make use of the simple relationship between the Spearman correlation and the parameter α shown in Figure 1 in a method of moments approach, it is not the case that Spearman is the best option. Figure 3  Examples in which more of the α's are non-zero will generally require iterative solutions of the moment equations, but except for that, they can be expected to yield reasonable estimates of the parameters provided that sample sizes are adequate, and enough moments are included in the estimation procedure. More complex examples exist for which a simple method of moments approach may be inadequate.

Example 3.
Consider the two-parameter case where α, β ∈ [ , ], α = αβ, α = α, and α = ( − α)β, and α = ( −αβ)( −α), and a sample, v, of size n, is available. This is a two-parameter subfamily which contains both the co-monotone copula and the counter-monotone copula, and therefore can exhibit the full range of correlations. It also, unsurprisingly, includes two rotated versions of the Ali-Mikhail-Haq copula. Approximate Bayesian Computation, or ABC, can be applied, where the assumed prior for (α, β) can be any distribution deemed appropriate. Here, we will assume it to be a bivariate distribution with independent Uniform( , ) marginals.